The
understanding of probability is complete when it relates not only
with
the concept’s definition, but also at the relationship between the
mathematical model and the real world of random processes. The
concept of probability, even we refer strictly to the
mathematical definition, is full of relativities and of
philosophical and psychological implications.
Relativity
of probability
When
we speak about the relativity of probability, we refer to the real
objective way in which probability theory models the hazard and in
which the human degree of belief in the occurrence of various events
is sufficiently theoretically justified to make decisions. Thus,
any criticism of the application of probability results in daily
life will not hint at the mathematical theory itself, but at the
transfer of theoretical information from the model to the
surrounding reality. Briefly, these relativities are:
1)
Conceptual relativities
a) Terminological
relativities
– Mathematical probability and
philosophical probability are different objects;
b) Relativities
of mathematical definition
– Defining a term through itself (the equally
possible attribute from the classical definition);
– The axiomatic nonstructural and
nonindividual definition of event (as an element of a collective
structure);
– Choosing the set of axioms
(Kolmogorov’s axiomatics in the complete definition);
2)
Relativities of equivalence of mathematical model with
real world
– The subject of philosophical
probability is hazard and randomness, which cannot be mathematized;
– Infinity, which is present in the
definition of mathematical concept of probability, is not found
again in the finite experimental reality;
– The event, as a unit of mathematical
theory, does not reproduce the event from the real world, which is
much more complex;
3)
Relativities of practical applications of probability
calculus
– Choosing the field of events;
– Idealizations of the equally
probable type;
– The subjective translation of the
result of the Law of Large Numbers for finite successions of
experiments.
These relativities require at least an additional circumspection of
the person who sees probability as an absolute degree of belief and
implicitly the limitation of making decisions based on the numerical
value of probability as a unique criterion.
Probability
may be simultaneously viewed as:
–
Limit of relative frequency within a sequence of tests
performed under theoretically identical conditions;
–
Objective measure of possibility;
–
Subjective degree of belief in the occurrence of an event.
There are also other interpretations of probability, resulting from
mathematical theories with similar structures:
–
Predicted relative frequency within a physical model
(Drieschner);
–
Measure of tendency of an experimental context to produce an
outcome (Popper);
–
Logical relation between a data field and a hypothesis with
respect to partial implication (Keynes);
–
Numerical expression of an information about the existence of
an event in certain conditions (Onicescu).
All these interpretations are characterized by logical equivalences
and contain elements having philosophical implications like
prediction, possibility, frequency and degree of belief.
Philosophy
of probability
What
is the sense of the question: “What is the probability of …”?
This seems to be the essential question around which all problems of
philosophy of probability revolve. Great mathematicians like Pascal,
Bernoulli, Laplace, Cornot, von Mises, Poincaré, Reichenbach,
Popper, de Finetti, Carnap and Onicescu performed philosophical
studies of the probability concept and dedicated to them an
important part of their research, but the major questions still
remain open to study:
● Can probability
also be defined in other terms besides through itself?
●
Can we verify that it exists, at least in principle? What
sense must be assigned to this existence? Does it express anything
besides a lack of knowledge?
●
Can a probability be assigned to an aleatory isolated event
or just to some collective structures?
These are just few of the basic questions that philosophy dealt
with, through the efforts of the thinkers listed earlier, but still
without a scientifically satisfactory conclusion. Hundreds of pages
of papers might be written on each of such kind of questions.
Probability has a double meaning: first as
a measure of the real possibility of things (the physical
probability revealed through frequency) and second as the degree of
trust; in other words, there exist a philosophical probability and a
mathematical one, and these are not to be confounded. The
probability of an event does not really exist in the phenomenal
world, like mass, force and the Greenwich meridian do not exist as
real objects. It only exists abstractly. Its objective significance
is that, starting from the same hypotheses, all mathematicians will
find the same value for it, no matter the individual subjective
opinions. It serves as a tool for acquiring a partial knowledge of
the surrounding world, which is not equivalently and totally
reproduced, simply because the hazard cannot be theoretically
modeled and quantified. And then what is the justification for
probability theory? What is the sense of its application? Humans are
sentenced to act in uncertainty conditions. If humans had an
infinite intelligence and calculus capacity, he or she could predict
the future and would know our entire past. Probability theory is the
mathematics of idealized hazard. Its application consists of
reducing all events of a certain type to an arbitrary number of
equally possible cases and calculating the number of favorable
cases. Probability is nothing more than the mathematical degree of
certainty we have about an event. It is simultaneously objective and
subjective. Probability does not exist beyond us. In fact, it is not
about the degree of certainty we have a priori, but one we should
have if we were perfectly rational and could make the equally
possible judgment. Therefore, probability is the only reasonable
way to behave in conditions of partial knowledge and uncertainty, by
using mathematics as a unique method, which is rigorous and
unanimously accepted.
Sources 
All
above subjects are touched on in the book UNDERSTANDING AND
CALCULATING THE ODDS: Probability Theory Basics and Calculus
Guide for Beginners, with Applications in Games of Chance and
Everyday Life. The
dedicated chapters
aim to stimulate the research and deep knowledge tendencies of
readers with regard to these subjects, to complete an image of
the probability concept that includes its philosophical and
psychological aspects and to extend the simple image of a
mathematical tool of calculus of degree of belief, which is so
common among average people. You
may find it in the Books
section with a free sample. 

